
As \(\Omega ^1(A)\)  is given by the quotient of \(\Omega ^1(k[x_1,\dots , x_n])\) 
by the ideal \((J_A,\mathrm {d}_x J_A)\)  generated by \(J_A\)  and \(\mathrm {d}_x J_A\) ,
and likewise \(\Omega ^1(B) = \Omega ^1(k[X_1,\dots , X_m])/(J_B,\mathrm {d}_XJ_B)\)  with
\(J_B := J_A\cap k[X_1,\dots ,X_m]\) , see, e.g., [1]},
the required isomorphism is induced by the following diagram with exact rows
\(@C=1.5em{0 [r] & (J_B,\mathrm {d}_XJ_B)[d]_-{\cong } [r] & \Omega ^1(k[X_1,\ldots , X_m]) [r][d]_-{\cong } & \Omega ^1(B)[r]@{-->}[d]^-{\cong } & 0 \\0 [r] & (J_B,\mathrm {d}_xJ_B) [r] & k[X_1,\ldots , X_m] \mathrm {d}_x\big (k[X_1,\ldots , X_m]\big ) [r] & B\,\mathrm {d}_A(B) [r] & 0 ,}\) 
